Seismic waves damping with arrays of inertial resonators

TL;DR

This paper explores the use of arrays of inertial resonators, specifically iron spheres connected via ligaments, to create seismic wave stop bands, potentially protecting infrastructure from low-frequency seismic waves.

Contribution

It introduces a novel design of inertial resonator arrays with specific ligament configurations to achieve seismic wave damping and stop bands at low frequencies.

Findings

Complete stop bands achieved in the 16-21 Hz range for large spheres.

Damping of elastodynamic waves from 8 to 49 Hz with optimized soil-connected spheres.

Bending modes are primarily responsible for seismic wave damping.

Abstract

We investigate the elastic stop band properties of a theoretical cubic array of iron spheres connected to a bulk of concrete via iron or rubber ligaments. Each sphere can move freely within a surrounding air cavity, but ligaments couple it to the bulk and further facilitate bending and rotational motions. Associated low frequency local resonances are well predicted by an asymptotic formula. We find complete stop bands (for all wave-polarizations) in the frequency range $[16,21]$ Hertz (resp. $[6,11]$ Hertz) for $7.4$-meter (resp. $0.74$-meter) diameter iron spheres with a $10$-meter (resp. $1$-meter) center-to-center spacing, when they are connected to concrete via steel (resp. rubber) ligaments. The scattering problem shows that only bending modes are responsible for damping and that rotational modes are totally overwritten by bending modes. Regarding seismic applications, we further consider soil as a bulk medium, in which case the relative bandwidth of the low frequency stop band can be enlarged through ligaments of different sizes that allow for well separated bending and rotational modes. We finally achieve some damping of elastodynamic waves from $8$ to $49$ Hertz (relative stop band of $143$ percent) for iron spheres $0.74$-meter in diameter that are connected to soil with six rubber ligaments of optimized shapes. These results represent a preliminary step in the design of seismic shields placed around, or underneath, the foundations of large civil infrastructures.